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Chapter 3: Eigenstructure, the Karhunen Loeve Transform, and Singular-Value Decomposition

R. Lynn Kirlin
R. Lynn Kirlin
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January 01, 1999

An M × M covariance matrix R exhibits many special properties. For example, it is complex Hermitian, equal to its conjugate transpose, RH = R; it is positive semidefinite, xHRx ≥ 0. Because of the latter, its eigenvalues are greater than or equal to zero as well. In many cases, it is also Toeplitz Ri,j = Ri + m,j + m, that is, diagonal elements are equal. In this chapter, I will review some of the more important properties of covariance matrices and their eigenstructure, and discuss some simple applications.

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Geophysical Developments Series

Covariance Analysis for Seismic Signal Processing

Society of Exploration Geophysicists
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Publication date:
January 01, 1999




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